**by Monte Basgall**

** The following is one in a series of articles on teaching at Duke.*

David Schaeffer thinks his unusual new spring course -- Mathematical Problems in Current Scientific and Engineering Research -- is an idea whose time has come.

Throughout last semester, the James B. Duke professor of mathematics and two math graduate students met with four graduate students in mechanical engineering and materials science and two of their faculty advisers to explore problems in advanced engineering research and theory.

"For my own personal sake, I have always been interested in seeing how mathematics related to other fields," Schaeffer said in an interview. "With this course, I can learn what some of my colleagues are doing in science that is new to me.

"I can also feel like I'm doing some good in making mathematical knowledge available to a wider audience. And another purpose is to teach our own mathematics graduate students how to talk to people in other fields. The job market is very rough, and employers are looking for applied skills and the ability to interact with other people.".

Last September, Schaeffer solicited possible class topics in a memorandum to Duke's science and engineering faculty. The memo requested themes in "an active area of research that would be useful for math students to learn." It also sought research "in which mathematics, either analytical or numerical, has a key contribution to make."

Schaeffer got "eight or nine" responses from faculty in the biological sciences, engineering, geology and physics, and "quite a few of these were very exciting indeed," he said. Out of those, he selected "shock wave lithotripsy."

Lithotripsy is one of several medical procedures that make use of high frequency sound, known as ultrasound. In this case, the ultrasound wave field induces tiny bubbles to form and then collapse inside the body, destroying kidney stones in the process.

The proposal was originally submitted by Laurens Howle, an assistant professor of mechanical engineering and materials science whose research interests, besides lithotripsy, also include "sonoluminescence" -- tiny flashes of light that are mysteriously emitted when ultrasound-induced bubbles collapse at tremendous pressures.

Pei Zhong, an associate research professor of mechanical engineering and materials science, joined the class and "added an invaluable perspective," Schaeffer said. Zhong is researching ways to make lithotripsy destroy stones more effectively while causing less trauma to surrounding tissues.

Both Howle and Zhong brought along graduate students. In what Schaeffer termed "a fairly conventional lecture format," students and faculty first reviewed what was known about the topic. Later, "we split up into two teams, one around each of the two professors," Schaeffer said.

Working with Zhong's engineering team, Schaeffer and his math students "isolated a mathematical model that was an extension of an existing model," Schaeffer added. "This is something that I expect to continue over at least the next few months and hopefully longer. It will surely result in a research publication and perhaps a fairly general model for how high intensity ultrasound can damage tissues."

In the case of Howle's team, "we had trouble isolating a specific problem, and then when we did, we found that what's going on seems to be very complex," Schaeffer said. "The difficulty is that there are several things going on at once, and it's hard to understand them all simultaneously. The way I think I was able to be helpful to them is that I asked questions from a different perspective, questions they might not have asked."

Both engineering professors expressed satisfaction with the experience.

"We have developed several computational models -- based on the course's contents -- that are helping us better understand the physical processes taking place in collapsing bubbles," said Howle, who attended with graduate students Mike Gustafson and Mark Murray. In fact, Howle predicted that the course "will spawn a significant portion of Mark's masters thesis and perhaps a portion of his Ph.D. dissertation."

Zhong, who brought along graduate students Iulian Cioanta and Henry Tong, added: "The interdisciplinary collaboration has been essential for us to gain physical insights into lithotripsy-induced cavitation (bubble collapse).

"The development of a general mathematical model for bubble activity in high-intensity ultrasound fields, initiated in this class, may have potential applications in shock wave lithotripsy, therapeutic ultrasound and drug delivery."

Although he was pleased with this year's experiences, Schaeffer said he will prepare for his next spring class a little differently. Next fall, after he solicits ideas from the science and engineering faculty, "I'll narrow down the selection to three or four and then ask those people to come and give a short summary of their work to the math graduate students. Then I will make the final choice in discussions with the graduate students."

This year, while one math graduate student "had a very good experience,"
Schaeffer thinks the other "was perhaps a little bewildered."
Ultimately, he said, "I want this to be of benefit to mathematicians.
I want to be contributing to my own department as well as to the university
at large."